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<p><dfn class="terminology">Convergence</dfn>  A power series <span class="process-math">\(\displaystyle\sum_{n=0}^{\infty} a_n(x - x_0)^n\)</span> <dfn class="terminology">converges</dfn> at a point <span class="process-math">\(x\)</span> if</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\lim_{m\to\infty}\sum_{n=0}^m a_n(x - x_0)^n
\end{equation*}
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<p class="continuation">exists for that <span class="process-math">\(x\text{.}\)</span>The series certainly converges for <span class="process-math">\(x = x_0\text{;}\)</span> it may converge for all <span class="process-math">\(x\text{,}\)</span> or it may converge for some <span class="process-math">\(x\)</span> and not for others.</p>
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